Regression Analysis: Avg. Tot. Score versus %Takers, T/S Ratio

Model Summary

S R-sq R-sq(adj) R-sq(pred)

34.2385 79.91% 79.06% 76.83%

Coefficients

Term Coef SE Coef T-Value P-Value VIF

Constant 1118.5 39.5 28.34 0.000

%Takers -2.547 0.187 -13.62 0.000 1.05

T/S Ratio -3.73 2.21 -1.69 0.098 1.05

1. What are the degrees of freedom associated with the t tests of the slopes?

A) 46

B) 47

C) 48

D) 49

E) 50

2.Under what conditions are the computed p-values for these tests valid?

3. Assuming those conditions are satisfied, which of the following gives a correct interpretation of the p-value of T/S Ratio?

A) At the 10% significance level, T/S Ratio is a significant linear predictor.

B) At the 10% significance level, T/S is a significant linear predictor in combination with %Takers.

C) At the 5% significance level, T/S Ratio is not a significant linear predictor in combination with %Takers.

D) Both b and c.

Please explain how you arrived to each of the answers. Thanks!

1. The degrees of freedom associated with a t-test of the slope for a regression analysis is calculated as (n - p - 1), where n is the number of observations and p is the number of predictor variables (excluding the constant term). From the given information, we can see that there are two predictor variables (%Takers and T/S Ratio), so the degrees of freedom would be (n - 2 - 1) = (n - 3). However, the number of observations (n) is not provided in the given information. Without the value of n, we cannot determine the exact degrees of freedom. Therefore, none of the options A), B), C), D), or E) is the correct answer.

2. The computed p-values for the t-tests of the slopes are valid under the following conditions:
a) The errors/residuals of the regression analysis are normally distributed.
b) The errors/residuals have constant variance (homoscedasticity).
c) The errors/residuals are independent.
d) The predictors are linearly independent of each other.
e) The data points are independent of each other.
f) The observations are randomly sampled from the population.

Without additional information, we cannot determine if these conditions are satisfied. Therefore, the correct answer is that we do not have enough information to determine the validity of the computed p-values.

3. From the given information, we can see that the p-value of T/S Ratio is 0.098. In order to interpret this p-value correctly, we need to compare it to the chosen significance level (alpha). From the given options, the two significance levels mentioned are 10% and 5%.

a) At the 10% significance level, T/S Ratio is a significant linear predictor. This statement is incorrect because the p-value of 0.098 is larger than 0.10 (10% significance level).

b) At the 10% significance level, T/S is a significant linear predictor in combination with %Takers. This statement is incorrect because it adds a condition (%Takers) that was not mentioned in the original question.

c) At the 5% significance level, T/S Ratio is not a significant linear predictor in combination with %Takers. This statement is also incorrect because the p-value of 0.098 is smaller than 0.05 (5% significance level).

d) Both b and c. Since both statements b and c are incorrect, this option cannot be the correct answer.

Therefore, there is no option provided that gives a correct interpretation of the p-value of T/S Ratio with the given information.